Relational Quadrilateralland. II. The Quantum Theory

TL;DR

This paper develops the quantum theory for the relational quadrilateral, introducing new mathematical frameworks and kinematical quantization distinctions, advancing the understanding of quantum relational models beyond the triangle case.

Contribution

It provides the first quantum treatment of the relational quadrilateral with novel mathematics and highlights the importance of kinematical quantization distinctions in relational quantum mechanics.

Findings

Distinct relational kinematical quantization for $ ext{CP}^2$

Application of perturbation theory and integrals in quantum cosmology

Extension of Machian and timeless approaches to quadrilaterals

Abstract

This paper provides the quantum treatment of the relational quadrilateral. The underlying reduced configuration spaces are $\mathbb{CP}^2$ and the cone over this, C($\mathbb{CP}^2$). We consider exact free and isotropic HO potential cases and perturbations about these. Moreover, our purely relational kinematical quantization is distinct from the usual one for $\mathbb{CP}^2$, which turns out to carry absolutist connotations instead. Thus this paper is the first to note absolute-versus-relational motion distinctions at the kinematical rather than dynamical level, and also an example of value to the discussion of kinematical quantization along the lines of Isham 1984. This treatment of the relational quadrilateral is the first relational QM with very new mathematics for a finite QM model, and is far more typical of the general quantum relational $N$-a-gon than the previously-studied case of the relational triangle. We consider useful integrals as regards perturbation theory and the peaking interpretation of quantum cosmology. We subsequently consider problem of time applications of this: quantum Kucha\v{r} beables, the Machian version of the semiclassical approach and the timeless naive Schrodinger interpretation. These go toward extending the combined Machian semiclassical-histories-timeless approach of [1] to the case of the quadrilateral, which will be treated in subsequent papers.